Download EC 131 - Perfectly competitive markets - firms

EC 131 - Perfectly competitive
markets - firms’ decisions
Boston College
Department of Economics
Inacio Guerberoff Lanari Bo
November 2011
In general, you will be given the firm’s cost structure and some demand information. For most arguments the cost structure will be enough. Throughout this handout,
consider the following cost structure:
T C = 200 + 3Q + 0.2Q2
M C = 3 + 0.4Q
Clearly, the firm faces a fixed cost of $200, since when Q = 0 T C = $200.
1 Short-run analysis
Given a price, we can easily derive the quantity produced. Suppose that P = $19. Since
the firm supply doesn’t change the market price, marginal revenue equals the price:
M R = P = 19
Profit maximization implies that M R = M C. Thus:
19 = 3 + 0.4Q
Q = 40
Thus, the firm will produce Q = 40. Will profits be positive? In order to see that,
remember that T R = P × Q. Thus:
T R = 19 × 40 = $760
The total costs can be found by simply replacing Q by 40:
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T C = 200 + 3 × 40 + 0.2 × (40)2 = $640
Profits are, thus:
P rof its = T R − T C = 760 − 640 = $120
The firm will, then, have positive profits of $120.
1.1 Shutdown price
Remember that the condition for shutdown is P < AV C. Remember also that T C =
F C + V C. It’s clear that the fixed cost (F C) is $200, and thus the variable cost V C is
given by:
V C = 3Q + 0.2Q2
and thus:
AV C =
3Q + 0.2Q2
VC
=
= 3 + 0.2Q
Q
Q
When we consider the fact that the profit maximization condition (M C = M R)
implies that the marginal cost curve is the supply curve, in order to know the minimal
shutdown price we need to find when is it that M C ≥ AV C:
M C ≥ AV C =⇒ 3 + 0.4Q ≥ 3 + 0.2Q =⇒ 0.4Q ≥ 0.2Q =⇒ 0.4 ≥ 0.2
Obviously that condition holds regardless of the value of Q. That is, M C ≥ AV C is
always true. Thus, as long as P is higher than the minimum value of AV C the firm will
produce a positive Q in the short-run. Since the minumum value of AV C is 3, then the
firm will shutdown only if P < $3.
2 Long-run analysis
In the long-run, firms will enter whenever profits are positive, and will exit whenever
they are negative, shifting prices down and up respectively, until economi profits are
zero. Since profits can be written as:
P rof its = Q × (P − AT C)
Then this implies that in the long-run P = AT C. This is, thus, the long-run zero
profit condition. The second condition we must use in order to specify the long-run
equilibrium is, of course, the profit maximization condition: M C = M R, where here
M R = P since it’s a perfectly competitive market. Putting both conditions together,
we have:
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P = AT C =⇒ P =
200 + 3Q + 0.2Q2
Q
P = M C =⇒ P = 3 + 0.4Q
Together:
200 + 3Q + 0.2Q2
= 3 + 0.4Q
Q
Which solves for:
√
Q = 10 10
Which is the long-run supply for that firm. In order to find the price, we can replace
back to any of the two expressions involving prices. Let’s use the M C curve:
√
P = M C =⇒ P = 3 + 0.4 × (10 10)
√
P = 3 + 4 10
√
√
Thus, in the long-run prices will be P = 3+4 10 and each firm will supply Q = 10 10.
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